Theorem dmdi2 32234

Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
dmdi2	⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))

Proof of Theorem dmdi2

Step	Hyp	Ref	Expression
1		dmdi 32232	. 2 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))
2		eqimss2 4038	. 2 ⊢ (((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))
3	1, 2	syl 17	1 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ∩ cin 3945   ⊆ wss 3946   class class class wbr 5145  (class class class)co 7416   C_ℋ cch 30859   ∨_ℋ chj 30863   𝑀_ℋ^* cdmd 30897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-iota 6498  df-fv 6554  df-ov 7419  df-dmd 32211

This theorem is referenced by: (None)